\(\int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx\) [177]

Optimal result

Integrand size = 13, antiderivative size = 71 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {3 x}{b^3}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4} \]

[Out]

3*x/b^3-1/2*x^3/b/arccoth(tanh(b*x+a))^2-3/2*x^2/b^2/arccoth(tanh(b*x+a))+3*(b*x-arccoth(tanh(b*x+a)))*ln(arcc
oth(tanh(b*x+a)))/b^4

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2199, 2189, 2188, 29} \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 x}{b^3} \]

[In]

Int[x^3/ArcCoth[Tanh[a + b*x]]^3,x]

[Out]

(3*x)/b^3 - x^3/(2*b*ArcCoth[Tanh[a + b*x]]^2) - (3*x^2)/(2*b^2*ArcCoth[Tanh[a + b*x]]) + (3*(b*x - ArcCoth[Ta
nh[a + b*x]])*Log[ArcCoth[Tanh[a + b*x]]])/b^4

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2188

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 2189

Int[(v_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[b*(x/a), x] - Dist[(b*u
- a*v)/a, Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 2199

Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[u^(m + 1)*(v^
n/(a*(m + 1))), x] - Dist[b*(n/(a*(m + 1))), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m
, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] &&  !(ILtQ[m + n, -2] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] &&  !IntegerQ[m]
) || (ILtQ[m, 0] &&  !IntegerQ[n]))

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))^2} \, dx}{2 b} \\ & = -\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {3 \int \frac {x}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2} \\ & = \frac {3 x}{b^3}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}-\frac {\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^3} \\ & = \frac {3 x}{b^3}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}-\frac {\left (3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^4} \\ & = \frac {3 x}{b^3}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=-\frac {b^3 x^3+3 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^3 \left (5+6 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )-b x \coth ^{-1}(\tanh (a+b x))^2 \left (11+6 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 b^4 \coth ^{-1}(\tanh (a+b x))^2} \]

[In]

Integrate[x^3/ArcCoth[Tanh[a + b*x]]^3,x]

[Out]

-1/2*(b^3*x^3 + 3*b^2*x^2*ArcCoth[Tanh[a + b*x]] + ArcCoth[Tanh[a + b*x]]^3*(5 + 6*Log[ArcCoth[Tanh[a + b*x]]]
) - b*x*ArcCoth[Tanh[a + b*x]]^2*(11 + 6*Log[ArcCoth[Tanh[a + b*x]]]))/(b^4*ArcCoth[Tanh[a + b*x]]^2)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.24 (sec) , antiderivative size = 4977, normalized size of antiderivative = 70.10

method	result	size
risch	\(\text {Expression too large to display}\)	\(4977\)

[In]

int(x^3/arccoth(tanh(b*x+a))^3,x,method=_RETURNVERBOSE)

[Out]

-2*I*(3*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-3*P
i*x^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+6*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1
))^3-6*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1))^2+3*Pi*x^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-6*Pi*x^2*csgn(
I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+3*Pi*x^2*csgn(I*exp(2*b*x+2*a))^3-3*Pi*x^2*csgn(I*exp(2*b*x+2*a))*csgn(
I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+3*Pi*x^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+12*I*x^2*ln(exp(b*
x+a))+6*Pi*x^2+4*I*x^3*b)/b^2/(Pi*csgn(I*exp(2*b*x+2*a))^3-2*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-Pi
*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2
*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+Pi*csgn(I*
exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2
+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2+4*I*ln(exp(b*x+a))+2*Pi)^2+3/b^3*x+3/2*I/
b^4*ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-Pi*csgn(
I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(
I/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*
x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi
*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi)*Pi*csgn(I/(exp(2*b
*x+2*a)+1))^3+3/4*I/b^4*ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b
*x+2*a)+1))-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2
*a)+1))^3-2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x
+a))*csgn(I*exp(2*b*x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp
(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi
)*Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-3/4*I/b^4*ln(
Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-Pi*csgn(I/(exp(
2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I/(exp(
2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))
^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I
*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi)*Pi*csgn(I/(exp(2*b*x+2*a)
+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+3/4*I/b^4*ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2
*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+
2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*
exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b
*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(ex
p(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi)*Pi*csgn(I*exp(2*b*x+2*a))^3-3/2*I/b^4*ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*cs
gn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+
2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b
*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-P
i*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+
1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi)*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-3/4*I/b^4*
ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-Pi*csgn(I/(e
xp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I/(e
xp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*
a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi*csg
n(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi)*Pi*csgn(I*exp(2*b*x+2*
a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+3/4*I/b^4*ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*
a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2
*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*e
xp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*
x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp
(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi)*Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+3/4*I/b^4*ln(Pi*csgn(I/(exp
(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-Pi*csgn(I/(exp(2*b*x+2*a)+1))
*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I/(exp(2*b*x+2*a)+1))
^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+Pi*csgn(I*e
xp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(2*b*x+2*a
)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi)*Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a
)+1))^3-3/2*I/b^4*ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a
)+1))-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1)
)^3-2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x+a))*c
sgn(I*exp(2*b*x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x
+2*a)+1))^2+Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi)*Pi*c
sgn(I/(exp(2*b*x+2*a)+1))^2+3/2*I/b^4*ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x
+2*a)/(exp(2*b*x+2*a)+1))-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(
I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*
csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2
*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*
a+4*I*b*x+2*Pi)*Pi+3/b^3*ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*
b*x+2*a)+1))-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+
2*a)+1))^3-2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*
x+a))*csgn(I*exp(2*b*x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(ex
p(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*P
i)*x-3/b^4*ln(Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-P
i*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*P
i*csgn(I/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp(b*x+a))*csgn(I*e
xp(2*b*x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1
))^2+Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+4*I*(ln(exp(b*x+a))-b*x-a)+4*I*a+4*I*b*x+2*Pi)*ln(exp(b*x+
a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.75 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {16 \, b^{3} x^{3} + 32 \, a b^{2} x^{2} - 32 \, a^{2} b x + 5 i \, \pi ^{3} + 2 \, \pi ^{2} {\left (4 \, b x + 15 \, a\right )} - 40 \, a^{3} + 4 i \, \pi {\left (4 \, b^{2} x^{2} - 8 \, a b x - 15 \, a^{2}\right )} - 6 \, {\left (8 \, a b^{2} x^{2} + 16 \, a^{2} b x - i \, \pi ^{3} - 2 \, \pi ^{2} {\left (2 \, b x + 3 \, a\right )} + 8 \, a^{3} + 4 i \, \pi {\left (b^{2} x^{2} + 4 \, a b x + 3 \, a^{2}\right )}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, {\left (4 \, b^{6} x^{2} + 8 \, a b^{5} x - \pi ^{2} b^{4} + 4 \, a^{2} b^{4} + 4 i \, \pi {\left (b^{5} x + a b^{4}\right )}\right )}} \]

[In]

integrate(x^3/arccoth(tanh(b*x+a))^3,x, algorithm="fricas")

[Out]

1/4*(16*b^3*x^3 + 32*a*b^2*x^2 - 32*a^2*b*x + 5*I*pi^3 + 2*pi^2*(4*b*x + 15*a) - 40*a^3 + 4*I*pi*(4*b^2*x^2 -
8*a*b*x - 15*a^2) - 6*(8*a*b^2*x^2 + 16*a^2*b*x - I*pi^3 - 2*pi^2*(2*b*x + 3*a) + 8*a^3 + 4*I*pi*(b^2*x^2 + 4*
a*b*x + 3*a^2))*log(I*pi + 2*b*x + 2*a))/(4*b^6*x^2 + 8*a*b^5*x - pi^2*b^4 + 4*a^2*b^4 + 4*I*pi*(b^5*x + a*b^4
))

Sympy [F]

\[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\int \frac {x^{3}}{\operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]

[In]

integrate(x**3/acoth(tanh(b*x+a))**3,x)

[Out]

Integral(x**3/acoth(tanh(a + b*x))**3, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.80 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.06 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {16 \, b^{3} x^{3} - 5 i \, \pi ^{3} + 30 \, \pi ^{2} a + 60 i \, \pi a^{2} - 40 \, a^{3} - 16 \, {\left (i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{2} + 8 \, {\left (\pi ^{2} b + 4 i \, \pi a b - 4 \, a^{2} b\right )} x}{4 \, {\left (4 \, b^{6} x^{2} - \pi ^{2} b^{4} - 4 i \, \pi a b^{4} + 4 \, a^{2} b^{4} - 4 \, {\left (i \, \pi b^{5} - 2 \, a b^{5}\right )} x\right )}} - \frac {3 \, {\left (-i \, \pi + 2 \, a\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{4}} \]

[In]

integrate(x^3/arccoth(tanh(b*x+a))^3,x, algorithm="maxima")

[Out]

1/4*(16*b^3*x^3 - 5*I*pi^3 + 30*pi^2*a + 60*I*pi*a^2 - 40*a^3 - 16*(I*pi*b^2 - 2*a*b^2)*x^2 + 8*(pi^2*b + 4*I*
pi*a*b - 4*a^2*b)*x)/(4*b^6*x^2 - pi^2*b^4 - 4*I*pi*a*b^4 + 4*a^2*b^4 - 4*(I*pi*b^5 - 2*a*b^5)*x) - 3/2*(-I*pi
+ 2*a)*log(-I*pi + 2*b*x + 2*a)/b^4

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.73 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {12 \, \pi ^{2} b x - 48 i \, \pi a b x - 48 \, a^{2} b x + 5 i \, \pi ^{3} + 30 \, \pi ^{2} a - 60 i \, \pi a^{2} - 40 \, a^{3}}{4 \, {\left (4 \, b^{6} x^{2} + 4 i \, \pi b^{5} x + 8 \, a b^{5} x - \pi ^{2} b^{4} + 4 i \, \pi a b^{4} + 4 \, a^{2} b^{4}\right )}} + \frac {x}{b^{3}} + \frac {3 \, {\left (-i \, \pi - 2 \, a\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{4}} \]

[In]

integrate(x^3/arccoth(tanh(b*x+a))^3,x, algorithm="giac")

[Out]

1/4*(12*pi^2*b*x - 48*I*pi*a*b*x - 48*a^2*b*x + 5*I*pi^3 + 30*pi^2*a - 60*I*pi*a^2 - 40*a^3)/(4*b^6*x^2 + 4*I*
pi*b^5*x + 8*a*b^5*x - pi^2*b^4 + 4*I*pi*a*b^4 + 4*a^2*b^4) + x/b^3 + 3/2*(-I*pi - 2*a)*log(I*pi + 2*b*x + 2*a
)/b^4

Mupad [B] (verification not implemented)

Time = 4.10 (sec) , antiderivative size = 620, normalized size of antiderivative = 8.73 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {x}{b^3}-\frac {x\,\left (3\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-12\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+12\,a^2\right )-\frac {5\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3-8\,a^3-6\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2+12\,a^2\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )}{4\,b}}{b^3\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2+x\,\left (8\,a\,b^4-4\,b^4\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )+4\,a^2\,b^3+4\,b^5\,x^2-4\,a\,b^3\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}+\frac {\ln \left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )\,\left (3\,\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-3\,\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+6\,b\,x\right )}{2\,b^4} \]

[In]

int(x^3/acoth(tanh(a + b*x))^3,x)

[Out]

x/b^3 - (x*(3*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)
) + 2*b*x)^2 - 12*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x
) - 1)) + 2*b*x) + 12*a^2) - (5*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2
*a)*exp(2*b*x) - 1)) + 2*b*x)^3 - 8*a^3 - 6*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) +
log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^2 + 12*a^2*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x)
- 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)))/(4*b))/(b^3*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*
exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^2 + x*(8*a*b^4 - 4*b^4*(2*a - log((2*exp(2*a)*ex
p(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)) + 4*a^2*b^3 + 4*b^5*x^2 - 4
*a*b^3*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b
*x)) + (log(log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) - log(-2/(exp(2*a)*exp(2*b*x) - 1)))*(3*log
(-2/(exp(2*a)*exp(2*b*x) - 1)) - 3*log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 6*b*x))/(2*b^4)